Depending on duty cycle of operation needs, you can judge what resistance of wire can fit the form, if you can't find a form design tool.
The thermal resistance of the wire will limit how much power it can handle with a a corresponding temperature rise and rate of rise thus determines maximum on time.
The power dissipation will be 12V^2/R and the thermal resistance will be much poorer than a heatsink but it has some mass, so it may be on the order of 30 deg'C per wat but have a 2 minute time constant. So if just using for seconds, that may be acceptable. A PTC polyfuse can protect it with the desired holding current.
Let's say you wanted 50 Watts for 10 seconds with 10% duty or 5Watts average. Then assuming my guestimates are reasonable, it might rise 150'C in a few minutes but ok for 10 seconds with a 30'C rise.
Thus for 50 Watts or 4 A@12Vdc you need to choose the AWG of wire and length to replace the existing volume of magnet wire with 3 Ohms worth of wire.
There are ratios often used to estimate the change in AWG to resistance per meter but I wont elaborate on that, you can use the AWG tables to convert your 3mm (AWG pairs to something like 0.5mm. You can use an Ohm meter or lookup in tables.
Basically to find ohms/meter, divide Volume Resistivity by area in circular mm^2.